Find the determinant of a particular matrix without a calculator.

Question

I would like to know the trick to solving this determinant problem.

Find the determinant of the matrix $$ \begin{bmatrix} 283&5&\pi&347.86\times10^{15^{83}}\\ 3136 & 56 & 5 & \cos(2.7402)\\ 6776 & 121 & 11 & 5\\ 2464 & 44 & 4 & 2 \end{bmatrix} $$ Hint: do not use a calculator.

This is a problem here (section 3.2). According to it, the answer is 6.

I have no idea how to do this. Obviously some of the entries in this matrix are red herrings, and I need to perform some trick. But the only manipulations I know towards computing the determinant is adding one row to another and multiplying a row by a scalar, and nothing in this matrix suggests that I do that.

I've tried decomposing the matrix as follows, $$ \begin{bmatrix} 283 & 5 & \pi & 347.86\times10^{15^{83}}\\ 56^2 & 56 & 5 & \cos(2.7402)\\ 11^2\cdot 56 & 11^2 & 11 & 5\\ 4\cdot11\cdot 56 & 4\cdot 11 & 4 & 2 \end{bmatrix},$$

though I'm not sure what I accomplished (I could also note that $56 = 14\cdot 4, 4 = 2^2$ etc. and do more decomposing). The problem is not all entries in any column/row are nice, and the nicest looking ones are the first and second column (or third and fourth row). Maybe the matrix is the product of two nice ones?

Answer 1

HINT: Note that $$ \begin{bmatrix}283 \\ 3136 \\ 6776 \\ 2464\end{bmatrix} - 56 \begin{bmatrix}5 \\ 56 \\ 121 \\ 44\end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ 0 \\ 0 \end{bmatrix} $$ which allows you to replace the leftmost column by $[3, 0, 0, 0]^T$. You should be able to then write the determinant of this $4\times4$ matrix in terms of the determinant of a smaller matrix.

This smaller matrix still doesn't have an immediately obvious determinant, but perhaps a similar trick will work on it.